CN105547326B - Gyro and Magnetic Sensor combined calibrating method - Google Patents

Gyro and Magnetic Sensor combined calibrating method Download PDF

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CN105547326B
CN105547326B CN201510901523.8A CN201510901523A CN105547326B CN 105547326 B CN105547326 B CN 105547326B CN 201510901523 A CN201510901523 A CN 201510901523A CN 105547326 B CN105547326 B CN 105547326B
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CN105547326A (en
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武元新
刘佩林
郁文贤
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Shanghai Jiaotong University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C25/00—Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C25/00—Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass
    • G01C25/005—Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass initial alignment, calibration or starting-up of inertial devices

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Abstract

A kind of gyro provided by the invention and Magnetic Sensor combined calibrating method, it is characterised in that comprise the following steps：Step 1, the correlation measurement model of Magnetic Sensor and gyro is obtained, includes calibrating parameters in the measurement model；Step 2, the determination value of the calibrating parameters is obtained using the constraint Nonlinear least squares fitting based on the calibrating parameters.Compared with prior art, beneficial effects of the present invention are as follows：1st, the misalignment that can have both demarcated between Magnetic Sensor and gyro, gyro zero bias can also be demarcated simultaneously；2nd, magnetic field is more stable during, and demarcation effect is better, and is not influenceed by any acceleration noise, therefore equipment remains stationary state is not needed in implementation process；3rd, it is registering with the posture of Inertial Measurement Unit (including gyro and accelerometer) available for Magnetic Sensor.

Description

Gyro and Magnetic Sensor combined calibrating method
Technical field
The present invention relates to sensor technical field, in particular to the combined calibrating side of a kind of gyro and Magnetic Sensor Method.
Background technology
Gyro and Magnetic Sensor (the latter also known as magnetometer, magnetometer) are frequently used for posture determination or scientific measurement field. The angular speed of gyro sensitive carrier, magnetometer sensitivity environmental magnetic field.When magnetometer is near ferromagnetic material, around magnetometer Magnetic field distorted, it is impossible to correct measurement goes out magnetic field intensity.Magnetic interference can be divided into two kinds of Hard Magnetic effect and soft magnetism effect.Firmly Magnetic effect is the additivity magnetic disturbance as caused by permanent magnet or electric current, and soft magnetism effect is induced by soft magnetic materials and produced, in background Soft magnetic materials in magnetic field, which can induce, produces the magnetic field of itself, and the intensity to background magnetic field and direction generation distortion.Except this Outside, because of manufacturing process imperfection, also there is constant multiplier, sensitive axes cross-couplings and biasing equal error in magnetometer, therefore, Before magnetometer, it is necessary to carry out calibration to above error.Demarcation mentioned here refers to demarcate inside magnetometer.
When magnetometer is when gyro is used together, it is necessary to carry out magnetometer extrinsic calibration, that is, need demarcate magnetometer with Coordinate system misalignment between gyro.Soft magnetism effect can not only cause the change of magnetometer inner parameter, also result in magnetometer Changed with the coordinate system misalignment of gyro.Therefore, before the use, it is necessary to carry out demarcation and magnetometer inside magnetometer Coordinate system misalignment demarcation between other sensors.Conventional magnetometer scaling method make use of the magnetic field intensity and magnetic of locality The unrelated such a fact of power instrument posture, advantage are not need external accessory, and shortcoming is can not to demarcate magnetometer and other The coordinate system misalignment of sensor.On the other hand, the zero offset error of inexpensive gyro (such as MEMS gyro) is larger, and makes every time Used time is all varied from, if not doing compensation directly uses gyro to measure value, will influence the coordinate system misalignment of magnetometer and gyro Footmark determines effect.
The content of the invention
For in the prior art the defects of, it is an object of the invention to provide it is a kind of solve above-mentioned technical problem gyro and magnetic Sensor combined calibrating method.
In order to solve the above technical problems, a kind of gyro provided by the invention and Magnetic Sensor combined calibrating method, including such as Lower step：
Step 1, the correlation measurement model of Magnetic Sensor and gyro is obtained, includes demarcation ginseng in the correlation measurement model Number；
Step 2, the calibrating parameters are obtained using the constraint Nonlinear least squares fitting based on the calibrating parameters It is determined that value.
Preferably, the calibrating parameters include coordinate system misalignment and gyro zero bias.
Preferably, the correlation measurement model is：
Wherein, mm(tk+1) represent+1 moment t of kthk+1When Magnetic Sensor coordinate system m under magnetic field vector, mm(tk) represent K-th of moment tkWhen Magnetic Sensor coordinate system m under magnetic field vector, k is positive integer, mm(t) represent that Magnetic Sensor is sat during moment t Magnetic field vector under mark system m,Magnetic Sensor and gyro coordinate system misalignment attitude matrix are represented,Represent top during moment t Spiral shell coordinate system b angular velocity vector, ε represent gyro zero bias vector；Vec () represents to play matrix according to the sequential concatenation of row Come；
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m.
Preferably, the constraint Nonlinear least squares fitting is exported by the correlation measurement model：
Wherein, S3Represent four dimensional vectors that mould is 1；λ is Lagrange coefficient；Q=[q0 q1 q2 q3]TFor Magnetic Sensor With gyro coordinate system misalignment attitude matrixCorresponding quaternary number, q0,q1,q2,q3Quaternary number q four components are represented respectively；εm The gyro zero bias under Magnetic Sensor coordinate system are represented,
Vec (C (q)) represents to get up Matrix C (q) according to the sequential concatenation of row；
The coefficient matrix W of Magnetic Sensor and gyro coordinate system misalignment attitude matrixkFor：
The coefficient matrix M of gyro zero biaskFor：
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m.
Preferably, Magnetic Sensor and gyro coordinate are obtained from the correlation measurement model using linear least square method It is the initial value of misalignment attitude matrixWith the initial value of gyro zero bias
Corresponding to being extracted from least square solutionWithWillOrthogonalization simultaneously utilizes attitude matrix and quaternary Several transformation relations obtains the initial value q of quaternary number(0), Lagrange coefficient initial value λ(0)It is set to zero.
Preferably, iterative calculation is until meet the condition of convergence：
Wherein, x(i+1)Represent parameter x to be estimated i+1 time iterative value, x(i)Represent parameter x to be estimated ith iteration value, J For Jacobian matrix derivative vector, H is Hessian matrix, and parameter x to be estimated is：
Wherein, the Jacobian matrix derivative vector J and the Hessian matrix H are：
Wherein,
Jλ=qTQ-1,
αk=Wkvec(C(q))+Mkεm-(mm(tk+1)-mm(tk)),
Compared with prior art, beneficial effects of the present invention are as follows：
1st, the misalignment that can have both demarcated between Magnetic Sensor and gyro, gyro zero bias can also be demarcated simultaneously；
2nd, magnetic field is more stable during, and demarcation effect is better, and is not influenceed by any acceleration noise, therefore implements During do not need equipment remains stationary state；
3rd, it is registering with the posture of Inertial Measurement Unit (including gyro and accelerometer) available for Magnetic Sensor.
Embodiment
With reference to specific embodiment, the present invention is described in detail.Following examples will be helpful to the technology of this area Personnel further understand the present invention, but the invention is not limited in any way.It should be pointed out that the ordinary skill to this area For personnel, without departing from the inventive concept of the premise, some changes and improvements can also be made.These belong to the present invention Protection domain.
In stabilizing magnetic field, the change of magnetic sensor measured value is entirely due to caused by the change of posture.It is based on This is true, and the invention provides the coordinate system misalignment between a kind of triaxial magnetometer and three axis accelerometer and the connection of gyro zero bias Close scaling method.Magnetometer is fixedly connected with gyro, fully the measurement of change posture and synchronous acquisition magnetometer and gyro.Magnetic force Instrument data can be used for inside magnetometer demarcating, and the data of magnetometer and gyro are provided commonly between the magnetometer and gyro of the present invention Coordinate system misalignment and gyro zero bias combined calibrating.Assume to have been realized in demarcating inside magnetometer below.
Gyro of the present invention includes with Magnetic Sensor combined calibrating method：Obtain the correlation measurement mould of Magnetic Sensor and gyro Type, the measurement model include the parameter such as coordinate system misalignment and gyro zero bias；Using the constraint based on the calibrating parameters Nonlinear least squares fitting obtains the calibrating parameters determination value.
Wherein, correlation measurement model is：
Wherein, mm(tk+1) represent+1 moment t of kthk+1When Magnetic Sensor coordinate system m under magnetic field vector, mm(tk) represent K-th of moment tkWhen Magnetic Sensor coordinate system m under magnetic field vector, k is positive integer, mm(t) represent that Magnetic Sensor is sat during moment t Magnetic field vector under mark system m,Magnetic Sensor and gyro coordinate system misalignment attitude matrix are represented,Represent top during moment t Spiral shell coordinate system b angular velocity vector, ε represent gyro zero bias vector；Vec () represents to play matrix according to the sequential concatenation of row Come；
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m.
Preferably, constraint Nonlinear least squares fitting is exported by correlation measurement model：
Wherein, S3Represent four dimensional vectors that mould is 1；λ is Lagrange coefficient；Q=[q0 q1 q2 q3]TFor Magnetic Sensor With gyro coordinate system misalignment attitude matrixCorresponding quaternary number, q0,q1,q2,q3Expression quaternary number q four components respectively, four First number q is expressed as q0+q1i+q2j+q3K, wherein, imaginary unit i, j, k meet operation rule：i0=j0=k0=1, i2=j2= k2=-1；εmThe gyro zero bias under Magnetic Sensor coordinate system are represented,
Vec (C (q)) represents to get up Matrix C (q) according to the sequential concatenation of row；
The coefficient matrix W of Magnetic Sensor and gyro coordinate system misalignment attitude matrixkFor：
The coefficient matrix M of gyro zero biaskFor：
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m.
Preferably, Magnetic Sensor is obtained from correlation measurement model using linear least square method to lose with gyro coordinate system The initial value of quasi- attitude matrixWith the initial value of gyro zero bias
Corresponding to being extracted from least square solutionWithWillOrthogonalization simultaneously utilizes attitude matrix and quaternary Several transformation relations obtains the initial value q of quaternary number(0), Lagrange coefficient initial value λ(0)It is set to zero.
Preferably, iterative calculation is until meet the condition of convergence：
Wherein, x(i+1)Represent parameter x to be estimated i+1 time iterative value, x(i)Represent parameter x to be estimated ith iteration value, J For Jacobian matrix derivative vector, H is Hessian matrix, and parameter x to be estimated is：
Wherein, Jacobian matrix derivative vector J and Hessian matrix H are：
Wherein,
Jλ=qTQ-1,
αk=Wkvec(C(q))+Mkεm-(mm(tk+1)-mm(tk)),
The specific embodiment of the present invention is described above.It is to be appreciated that the invention is not limited in above-mentioned Particular implementation, those skilled in the art can make a variety of changes or change within the scope of the claims, this not shadow Ring the substantive content of the present invention.In the case where not conflicting, the feature in embodiments herein and embodiment can any phase Mutually combination.

Claims (5)

1. a kind of gyro and Magnetic Sensor combined calibrating method, it is characterised in that comprise the following steps：
Step 1, the correlation measurement model of Magnetic Sensor and gyro is obtained, includes calibrating parameters in the correlation measurement model； The correlation measurement model is：
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Wherein, mm(tk+1) represent+1 moment t of kthk+1When Magnetic Sensor coordinate system m under magnetic field vector, mm(tk) represent kth Individual moment tkWhen Magnetic Sensor coordinate system m under magnetic field vector, k is positive integer, mm(t) Magnetic Sensor coordinate system during moment t is represented Magnetic field vector under m,Magnetic Sensor and gyro coordinate system misalignment attitude matrix are represented,Represent that gyro is sat during moment t Mark system b angular velocity vector, ε represent gyro zero bias vector；Vec () represents to get up matrix according to the sequential concatenation of row；
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m；
Step 2, the determination of the calibrating parameters is obtained using the constraint Nonlinear least squares fitting based on the calibrating parameters Value.
2. gyro according to claim 1 and Magnetic Sensor combined calibrating method, it is characterised in that the calibrating parameters bag Include coordinate system misalignment and gyro zero bias.
3. gyro according to claim 1 and Magnetic Sensor combined calibrating method, it is characterised in that the constraint is non-linear Least-squares estimation is exported by the correlation measurement model：
<mrow> <munder> <mi>min</mi> <mrow> <mi>q</mi> <mo>&amp;Element;</mo> <msup> <mi>S</mi> <mn>3</mn> </msup> <mo>,</mo> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> </munder> <mo>{</mo> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <mo>|</mo> <mo>|</mo> <msub> <mi>W</mi> <mi>k</mi> </msub> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>C</mi> <mo>(</mo> <mi>q</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>M</mi> <mi>k</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>}</mo> <mo>+</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msup> <mi>q</mi> <mi>T</mi> </msup> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein, S3Represent four dimensional vectors that mould is 1；λ is Lagrange coefficient；Q=[q0 q1 q2 q3]TFor Magnetic Sensor and top Spiral shell coordinate system misalignment attitude matrixCorresponding quaternary number, q0,q1,q2,q3Quaternary number q four components are represented respectively；εmRepresent Gyro zero bias under Magnetic Sensor coordinate system,
<mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>q</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>q</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>3</mn> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>q</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>q</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>3</mn> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>q</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>q</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>q</mi> <mn>3</mn> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
Vec (C (q)) represents to get up Matrix C (q) according to the sequential concatenation of row；
The coefficient matrix W of Magnetic Sensor and gyro coordinate system misalignment attitude matrixkFor：
<mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msubsup> <msubsup> <mi>&amp;omega;</mi> <mrow> <mi>i</mi> <mi>b</mi> </mrow> <mrow> <mi>b</mi> <mi>T</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>&amp;times;</mo> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>;</mo> </mrow>
The coefficient matrix M of gyro zero biaskFor：
<mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msubsup> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>&amp;times;</mo> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>;</mo> </mrow>
mm(t) × represent by three-dimensional vector mm(t)=[mm(t)1 mm(t)2 mm(t)3]TThe multiplication cross matrix of composition, i.e.,Wherein mm(t)1Represent the magnetic under Magnetic Sensor coordinate system m during moment t Component in the X-direction of field vector；Wherein mm(t)2Represent the Y-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m On component；Wherein mm(t)3Represent the component in the Z-direction of magnetic field vector during moment t under Magnetic Sensor coordinate system m.
4. gyro according to claim 3 and Magnetic Sensor combined calibrating method, it is characterised in that utilize a linear most young waiter in a wineshop or an inn Multiply the initial value that method obtains Magnetic Sensor and gyro coordinate system misalignment attitude matrix from the correlation measurement modelWith The initial value of gyro zero bias
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mrow> <mo>(</mo> <msubsup> <mi>C</mi> <mi>b</mi> <mi>m</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mrow> <mo>(</mo> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>W</mi> <mi>k</mi> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>W</mi> <mi>k</mi> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>W</mi> <mi>k</mi> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>-</mo> <msup> <mi>m</mi> <mi>m</mi> </msup> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Corresponding to being extracted from least square solutionWithWillOrthogonalization simultaneously utilizes attitude matrix and quaternary number Transformation relation obtains the initial value q of quaternary number(0), Lagrange coefficient initial value λ(0)It is set to zero.
5. gyro according to claim 4 and Magnetic Sensor combined calibrating method, it is characterised in that iterative calculation is until full The sufficient condition of convergence：
<mrow> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msup> <mo>-</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mi>H</mi> <msub> <mo>|</mo> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msup> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>J</mi> <msub> <mo>|</mo> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msup> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>;</mo> </mrow>
Wherein, x(i+1)Represent parameter x to be estimated i+1 time iterative value, x(i)Parameter x to be estimated ith iteration value is represented, J is refined Than matrix derivative vector, H is Hessian matrix, and parameter x to be estimated is：
<mrow> <mi>x</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mi>q</mi> <mi>T</mi> </msup> </mtd> <mtd> <msubsup> <mi>&amp;epsiv;</mi> <mi>m</mi> <mi>T</mi> </msubsup> </mtd> <mtd> <mi>&amp;lambda;</mi> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>;</mo> </mrow>
Wherein, the Jacobian matrix derivative vector J and the Hessian matrix H are：
<mrow> <mi>J</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>J</mi> <mi>q</mi> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>J</mi> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> <mi>T</mi> </msubsup> </mtd> <mtd> <msub> <mi>J</mi> <mi>&amp;lambda;</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow>
<mrow> <mi>H</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>H</mi> <mrow> <mi>q</mi> <mi>q</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>H</mi> <mrow> <msub> <mi>q&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>H</mi> <mrow> <mi>q</mi> <mi>&amp;lambda;</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>H</mi> <mrow> <msub> <mi>q&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> <mi>T</mi> </msubsup> </mtd> <mtd> <msub> <mi>H</mi> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mo>&amp;times;</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>H</mi> <mrow> <mi>q</mi> <mi>&amp;lambda;</mi> </mrow> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mn>0</mn> <mrow> <mn>3</mn> <mo>&amp;times;</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
Wherein,
<mrow> <msub> <mi>J</mi> <mi>q</mi> </msub> <mo>=</mo> <mn>2</mn> <mi>&amp;lambda;</mi> <mi>q</mi> <mo>+</mo> <mn>2</mn> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msup> <mo>&amp;part;</mo> <mi>T</mi> </msup> <mi>q</mi> <mrow> <mo>(</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>J</mi> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> </msub> <mo>=</mo> <mn>2</mn> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msubsup> <mi>M</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>J</mi> <mi>&amp;lambda;</mi> </msub> <mo>=</mo> <msup> <mi>q</mi> <mi>T</mi> </msup> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mrow> <mi>q</mi> <mi>q</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>&amp;lambda;I</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>2</mn> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <mo>{</mo> <msubsup> <mo>&amp;part;</mo> <mi>q</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>W</mi> <mi>k</mi> </msub> <msub> <mo>&amp;part;</mo> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>Q</mi> <mn>0</mn> <mi>T</mi> </msubsup> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>Q</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>Q</mi> <mn>2</mn> <mi>T</mi> </msubsup> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>Q</mi> <mn>3</mn> <mi>T</mi> </msubsup> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>}</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <msub> <mi>H</mi> <mrow> <msub> <mi>q&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <mn>2</mn> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msup> <mo>&amp;part;</mo> <mi>T</mi> </msup> <mi>q</mi> <mrow> <mo>(</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <msubsup> <mi>W</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>M</mi> <mi>k</mi> </msub> <mo>,</mo> </mrow>
<mrow> <msub> <mi>H</mi> <mrow> <mi>q</mi> <mi>&amp;lambda;</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>q</mi> <mo>,</mo> <msub> <mi>H</mi> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <mn>2</mn> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msubsup> <mi>M</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>M</mi> <mi>k</mi> </msub> </mrow>
αk=Wkvec(C(q))+Mkεm-(mm(tk+1)-mm(tk)),
<mrow> <msub> <mo>&amp;part;</mo> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mi>e</mi> <mi>c</mi> <mo>(</mo> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>q</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mn>0</mn> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>q</mi> <mn>3</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
<mrow> <msub> <mi>Q</mi> <mn>0</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>Q</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>Q</mi> <mn>3</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
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